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On second order conditions in the multivariate block maxima and peak over threshold method

Author

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  • Bücher, Axel
  • Volgushev, Stanislav
  • Zou, Nan

Abstract

Second order conditions provide a natural framework for establishing asymptotic results about estimators for tail related quantities. Such conditions are typically tailored to the estimation principle at hand, and may be vastly different for estimators based on the block maxima (BM) method or the peak-over-threshold (POT) approach. In this paper we provide details on the relationship between typical second order conditions for BM and POT methods in the multivariate case. We show that the two conditions typically imply each other, but with a possibly different second order parameter. The latter implies that, depending on the data generating process, one of the two methods can attain faster convergence rates than the other. The class of multivariate Archimax copulas is examined in detail; we find that this class contains models for which the second order parameter is smaller for the BM method and vice versa. The theory is illustrated by a small simulation study.

Suggested Citation

  • Bücher, Axel & Volgushev, Stanislav & Zou, Nan, 2019. "On second order conditions in the multivariate block maxima and peak over threshold method," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 604-619.
    • Handle: RePEc:eee:jmvana:v:173:y:2019:i:c:p:604-619
    DOI: 10.1016/j.jmva.2019.04.011
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    References listed on IDEAS

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    1. Rafael Schmidt & Ulrich Stadtmüller, 2006. "Non‐parametric Estimation of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 307-335, June.
    2. Einmahl, J.H.J. & Krajina, A. & Segers, J., 2011. "An M-Estimator for Tail Dependence in Arbitrary Dimensions," Discussion Paper 2011-013, Tilburg University, Center for Economic Research.
    3. Drees, Holger & Huang, Xin, 1998. "Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 25-47, January.
    4. Bucher, Axel & Segers, Johan, 2014. "Extreme value copula estimation based on block maxima of a multivariate stationary time series," LIDAM Reprints ISBA 2014019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Charpentier, Arthur & Segers, Johan, 2009. "Tails of multivariate Archimedean copulas," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1521-1537, August.
    6. Drees, Holger & de Haan, Laurens & Li, Deyuan, 2003. "On large deviation for extremes," Statistics & Probability Letters, Elsevier, vol. 64(1), pages 51-62, August.
    7. Einmahl, J.H.J. & Segers, J.J.J., 2008. "Maximum Empirical Likelihood Estimation of the Spectral Measure of an Extreme Value Distribution," Other publications TiSEM e9340b9a-fe69-4e77-8594-8, Tilburg University, School of Economics and Management.
    8. Philippe Naveau & Armelle Guillou & Daniel Cooley & Jean Diebolt, 2009. "Modelling pairwise dependence of maxima in space," Biometrika, Biometrika Trust, vol. 96(1), pages 1-17.
    9. Bucher, Axel & Segers, Johan, 2018. "Maximum likelihood estimation for the Frechet distribution based on block maxima extracted from a time series," LIDAM Reprints ISBA 2018001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Charpentier, A. & Fougères, A.-L. & Genest, C. & Nešlehová, J.G., 2014. "Multivariate Archimax copulas," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 118-136.
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